LINEAR EQUATIONS
Linear equations [ edit ] Main articles: Linear equation , Linear differential equation , and Systems of linear equations Systems of homogeneous linear equations are closely tied to vector spaces. [14] For example, the solutions of � + 3 � + � = 0 4 � + 2 � + 2 � = 0 are given by triples with arbitrary � , � = � / 2 , and � = − 5 � / 2. They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. Matrices can be used to condense multiple linear equations as above into one vector equation, namely � � = 0 , where � = [ 1 3 1 4 2 2 ] is the matrix containing the coefficients of the given equations, � is the vector ( � , � , � ) , � � denotes the matrix product , and 0 = ( 0 , 0 ) is the zero vector. In a similar vein, the solutions of homogeneous lin...